QuestionA polynomial has a zero at $x=4 i$ . Which expression is a factor? a. $(x^{2}+8)$ b. $(x^{2}-8)$ c. $(x^{2}+16)$ d. $(x^{2}-16)$

Studdy Solution

STEP 1

Assumptions1. The polynomial function has a zero at $x=4i$ . . The polynomial function is a complex function, which means it has complex coefficients.
3. According to the Fundamental Theorem of Algebra, if a polynomial function has a complex root, then its conjugate is also a root.

STEP 2

We know that if $4i$ is a root, then its conjugate $-4i$ is also a root. We can express these roots as factors of the polynomial.
$x -4i =0$ $x +4i =0$

STEP 3

We can multiply these factors together to get a quadratic expression that must be a factor of the polynomial.
$(x -i)(x +i)$

STEP 4

Use the difference of squares formula, $a^2 - b^2 = (a - b)(a + b)$ , to simplify the expression.
$(x -4i)(x +4i) = x^2 - (4i)^2$

STEP 5

Calculate the square of $4i$ .
$(4i)^2 = -16$

STEP 6

Substitute $-16$ back into the expression.
$x^2 - (4i)^2 = x^2 - (-16) = x^2 +16$ The expression $x^2 +16$ must be a factor of the polynomial. Therefore, the correct answer is option c. $\left(x^{2}+16\right)$ .

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QuestionA polynomial has a zero at $x=4 i$ . Which expression is a factor? a. $(x^{2}+8)$ b. $(x^{2}-8)$ c. $(x^{2}+16)$ d. $(x^{2}-16)$

Studdy Solution

STEP 1

Assumptions1. The polynomial function has a zero at $x=4i$ . . The polynomial function is a complex function, which means it has complex coefficients.
3. According to the Fundamental Theorem of Algebra, if a polynomial function has a complex root, then its conjugate is also a root.

STEP 2

We know that if $4i$ is a root, then its conjugate $-4i$ is also a root. We can express these roots as factors of the polynomial.
$x -4i =0$ $x +4i =0$

STEP 3

We can multiply these factors together to get a quadratic expression that must be a factor of the polynomial.
$(x -i)(x +i)$

STEP 4

Use the difference of squares formula, $a^2 - b^2 = (a - b)(a + b)$ , to simplify the expression.
$(x -4i)(x +4i) = x^2 - (4i)^2$

STEP 5

Calculate the square of $4i$ .
$(4i)^2 = -16$

STEP 6

Substitute $-16$ back into the expression.
$x^2 - (4i)^2 = x^2 - (-16) = x^2 +16$ The expression $x^2 +16$ must be a factor of the polynomial. Therefore, the correct answer is option c. $\left(x^{2}+16\right)$ .

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QuestionA polynomial has a zero at x=4ix=4 ix=4i. Which expression is a factor? a. (x2+8)(x^{2}+8)(x2+8) b. (x2−8)(x^{2}-8)(x2−8) c. (x2+16)(x^{2}+16)(x2+16) d. (x2−16)(x^{2}-16)(x2−16)

Studdy Solution

STEP 1

Assumptions1. The polynomial function has a zero at x=4ix=4ix=4i. . The polynomial function is a complex function, which means it has complex coefficients.3. According to the Fundamental Theorem of Algebra, if a polynomial function has a complex root, then its conjugate is also a root.

STEP 2

We know that if 4i4i4i is a root, then its conjugate −4i-4i−4i is also a root. We can express these roots as factors of the polynomial.x−4i=0x -4i =0x−4i=0x+4i=0x +4i =0x+4i=0

STEP 3

We can multiply these factors together to get a quadratic expression that must be a factor of the polynomial.(x−i)(x+i)(x -i)(x +i)(x−i)(x+i)

STEP 4

Use the difference of squares formula, a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)a2−b2=(a−b)(a+b), to simplify the expression.(x−4i)(x+4i)=x2−(4i)2(x -4i)(x +4i) = x^2 - (4i)^2(x−4i)(x+4i)=x2−(4i)2

STEP 5

Calculate the square of 4i4i4i.(4i)2=−16(4i)^2 = -16(4i)2=−16

STEP 6

Substitute −16-16−16 back into the expression.x2−(4i)2=x2−(−16)=x2+16x^2 - (4i)^2 = x^2 - (-16) = x^2 +16x2−(4i)2=x2−(−16)=x2+16The expression x2+16x^2 +16x2+16 must be a factor of the polynomial. Therefore, the correct answer is option c. (x2+16)\left(x^{2}+16\right)(x2+16).

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QuestionA polynomial has a zero at $x=4 i$ . Which expression is a factor? a. $(x^{2}+8)$ b. $(x^{2}-8)$ c. $(x^{2}+16)$ d. $(x^{2}-16)$

Assumptions1. The polynomial function has a zero at $x=4i$ . . The polynomial function is a complex function, which means it has complex coefficients.
3. According to the Fundamental Theorem of Algebra, if a polynomial function has a complex root, then its conjugate is also a root.

We know that if $4i$ is a root, then its conjugate $-4i$ is also a root. We can express these roots as factors of the polynomial.
$x -4i =0$ $x +4i =0$

We can multiply these factors together to get a quadratic expression that must be a factor of the polynomial.
$(x -i)(x +i)$

Use the difference of squares formula, $a^2 - b^2 = (a - b)(a + b)$ , to simplify the expression.
$(x -4i)(x +4i) = x^2 - (4i)^2$

Calculate the square of $4i$ .
$(4i)^2 = -16$

Substitute $-16$ back into the expression.
$x^2 - (4i)^2 = x^2 - (-16) = x^2 +16$ The expression $x^2 +16$ must be a factor of the polynomial. Therefore, the correct answer is option c. $\left(x^{2}+16\right)$ .